A circle has a radius of $7$. An arc in this circle has a central angle of $\dfrac{6}{5}\pi$ radians. What is the length of the arc? ${14\pi}$ ${\dfrac{6}{5}\pi}$ $\color{#DF0030}{\dfrac{42}{5}\pi}$ ${7}$
Solution: First, calculate the circumference of the circle. $c = 2\pi r = 2\pi (7) = 14\pi$ The ratio between the arc's central angle $\theta$ and $2 \pi$ radians is equal to the the ratio between the arc length $s$ and the circle's circumference $c$ $\dfrac{\theta}{2 \pi} = \dfrac{s}{c}$ $\dfrac{6}{5}\pi \div 2 \pi = \dfrac{s}{14\pi}$ $\dfrac{3}{5} = \dfrac{s}{14\pi}$ $\dfrac{3}{5} \times 14\pi = s$ $\dfrac{42}{5}\pi = s$